Econ 240C Lecture 16

2 Part I. VAR Does the Federal Funds Rate Affect Capacity Utilization?

3 The Federal Funds Rate is one of the principal monetary instruments of the Federal Reserve Does it affect the economy in “real terms”, as measured by capacity utilization for total industry?

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6 Preliminary Analysis The Time Series, Monthly, January 1967 through April 2008

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8 Capacity Utilization Total Industry: Jan April 2008

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12 Identification of TCU Trace Histogram Correlogram Unit root test Conclusion: probably evolutionary

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17 Identification of FFR Trace Histogram Correlogram Unit root test Conclusion: unit root

18 Pre-whiten both

19 Changes in FFR & Capacity Utilization

20 Contemporaneous Correlation

21 Dynamics: Cross-correlation Two-Way Causality?

22 In Levels Too much structure in each hides the relationship between them

23 In differences

24 Granger Causality: Four Lags

25 Granger Causality: two lags

26 Granger Causality: Twelve lags

27 Estimate VAR

28 Estimation of VAR

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35 Specification Same number of lags in both equations Use liklihood ratio tests to compare 12 lags versus 24 lags for example

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37 Estimation Results OLS Estimation each series is positively autocorrelated –lags 1, 18 and 24 for dtcu – lags 1, 2, 4, 7, 8, 9, 13, 16 for dffr each series depends on the other –dtcu on dffr: negatively at lags 10, 12, 17, 21 –dffr on dtcu: positively at lags 1, 2, 9, 24 and negatively at lag 12

38 We Have Mutual Causality, But We Already Knew That DTCU DFFR

39 Correlogram of DFFR

40 Correlogram of DTCU

41 Interpretation We need help Rely on assumptions

42 What If What if there were a pure shock to dtcu –as in the primitive VAR, a shock that only affects dtcu immediately

43 Primitive VAR (tcu Notation) dtcu(t) =  1 +  1 dffr(t) +  11 dtcu(t-1) +  12 dffr(t-1) +  1 x(t) + e dtcu (t) (2) dffr(t) =  2 +  2 dtcu(t) +  21 dtcu(t-1) +  22 dffr(t-1) +  2 x(t) + e dffr (t)

Primitive VAR (capu notation)

45 The Logic of What If A shock, e dffr, to dffr affects dffr immediately, but if dcapu depends contemporaneously on dffr, then this shock will affect it immediately too so assume   is zero, then dcapu depends only on its own shock, e dcapu, first period But we are not dealing with the primitive, but have substituted out for the contemporaneous terms Consequently, the errors are no longer pure but have to be assumed pure

46 DTCU DFFR shock

47 Standard VAR dcapu(t) = (        /(1-     ) +[ (    +     )/(1-     )] dcapu(t-1) + [ (    +     )/(1-     )] dffr(t-1) + [(    +      (1-     )] x(t) + (e dcapu  (t) +   e dffr  (t))/(1-     ) But if we assume    then  dcapu(t) =    +    dcapu(t-1) +   dffr(t-1) +    x(t) + e dcapu  (t) + 

48 Note that dffr still depends on both shocks dffr(t) = (        /(1-     ) +[(      +   )/(1-     )] dcapu(t-1) + [ (      +   )/(1-     )] dffr(t- 1) + [(      +    (1-     )] x(t) + (    e dcapu  (t) + e dffr  (t))/(1-     ) dffr(t) = (        +[(      +   ) dcapu(t-1) + (      +   ) dffr(t-1) + (      +    x(t) + (    e dcapu (t) + e dffr  (t))

49 DTCU DFFR shock e dtcu  (t) e dffr  (t) Reality

50 DTCU DFFR shock e dtcu  (t) e dffr  (t) What If

51 EVIEWS

52 Economy affects Fed, not vice versa

53 Interpretations Response of dtcu to a shock in dtcu –immediate and positive: autoregressive nature Response of dffr to a shock in dffr –immediate and positive: autoregressive nature Response of dtcu to a shock in dffr –starts at zero by assumption that    –interpret as Fed having no impact on TCU Response of dffr to a shock in dtcu –positive and then damps out –interpret as Fed raising FFR if TCU rises

54 Change the Assumption Around

55 DTCU DFFR shock e dtcu  (t) e dffr  (t) What If

56 Standard VAR dffr(t) = (        /(1-     ) +[(      +   )/(1-     )] dcapu(t-1) + [ (      +   )/(1-     )] dffr(t- 1) + [(      +    (1-     )] x(t) + (    e dcapu  (t) + e dffr  (t))/(1-     ) if    then, dffr(t) =      dcapu(t-1) +   dffr(t-1) +    x(t) + e dffr  (t)) but, dcapu(t) = (        + (    +     ) dcapu(t- 1) + [ (    +     ) dffr(t-1) + [(    +      x(t) + (e dcapu  (t) +   e dffr  (t))

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58 Interpretations Response of dtcu to a shock in dtcu –immediate and positive: autoregressive nature Response of dffr to a shock in dffr –immediate and positive: autoregressive nature Response of dtcu to a shock in dffr –is positive (not - ) initially but then damps to zero –interpret as Fed having no or little control of TCU Response of dffr to a shock in dtcu –starts at zero by assumption that    –interpret as Fed raising FFR if CAPU rises

59 Conclusions We come to the same model interpretation and policy conclusions no matter what the ordering, i.e. no matter which assumption we use,    or    So, accept the analysis

60 Understanding through Simulation We can not get back to the primitive fron the standard VAR, so we might as well simplify notation y(t) = (        /(1-     ) +[ (    +     )/(1-     )] y(t-1) + [ (    +     )/(1-     )] w(t-1) + [(   +      (1-     )] x(t) + (e dcapu  (t) +   e dffr  (t))/(1-     ) becomes y(t) = a 1 + b 11 y(t-1) + c 11 w(t-1) + d 1 x(t) + e 1 (t)

61 And w(t) = (        /(1-     ) +[(     +   )/(1-     )] y(t-1) + [ (      +   )/(1-     )] w(t-1) + [(      +    (1-     )] x(t) + (    e dcapu  (t) + e dffr  (t))/(1-     ) becomes w(t) = a 2 + b 21 y(t-1) + c 21 w(t-1) + d 2 x(t) + e 2 (t)

62 Numerical Example y(t) = 0.7 y(t-1) w(t-1)+ e 1 (t) w(t) = 0.2 y(t-1) w(t-1) + e 2 (t) where e 1 (t) = e y  (t) e w  (t) e 2 (t) = e w  (t)

63 Generate e y (t) and e w (t) as white noise processes using nrnd and where e y (t) and e w (t) are independent. Scale e y (t) so that the variances of e 1 (t) and e 2 (t) are equal –e y (t) = 0.6 *nrnd and –e w (t) = nrnd (different nrnd) Note the correlation of e 1 (t) and e 2 (t) is 0.8

64 Analytical Solution Is Possible These numerical equations for y(t) and w(t) could be solved for y(t) as a distributed lag of e 1 (t) and a distributed lag of e 2 (t), or, equivalently, as a distributed lag of e y (t) and a distributed lag of e w (t) However, this is an example where simulation is easier

65 Simulated Errors e 1 (t) and e 2 (t)

66 Simulated Errors e 1 (t) and e 2 (t)

67 Estimated Model

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73 Y to shock in w Calculated

Impact of shock in w on variable y